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About this book

This Festschrift is dedicated to Götz Trenkler on the occasion of his 65th birthday. As can be seen from the long list of contributions, Götz has had and still has an enormous range of interests, and colleagues to share these interests with. He is a leading expert in linear models with a particular focus on matrix algebra in its relation to statistics. He has published in almost all major statistics and matrix theory journals. His research activities also include other areas (like nonparametrics, statistics and sports, combination of forecasts and magic squares, just to mention afew). Götz Trenkler was born in Dresden in 1943. After his school years in East G- many and West-Berlin, he obtained a Diploma in Mathematics from Free University of Berlin (1970), where he also discovered his interest in Mathematical Statistics. In 1973, he completed his Ph.D. with a thesis titled: On a distance-generating fu- tion of probability measures. He then moved on to the University of Hannover to become Lecturer and to write a habilitation-thesis (submitted 1979) on alternatives to the Ordinary Least Squares estimator in the Linear Regression Model, a topic that would become his predominant ?eld of research in the years to come.

Table of Contents


Nonparametric Inference

Adaptive Tests for the c-Sample Location Problem

This paper deals with the concept of adaptive tests and with an application to the c-sample location problem. Parametric tests like the ANOVA F-tests are based on the assumption of normality of the data which is often violated in practice. In general, the practising statistician has no clear idea of the underlying distribution of his data. Thus, an adaptive test should be applied which takes into account the given data set. We use the concept of Hogg [21], i.e. to classify, at first, the unknown distribution function with respect to two measures, one for skewness and one for tailweight, and then, at the second stage, to select an appropriate test for that classified type of distribution. It will be shown that under certain conditions such a two-staged adaptive test maintains the level. Meanwhile, there are a lot of proposals for adaptive tests in the literature in various statistical hypotheses settings. It turns out that all these adaptive tests are very efficient over a broad class of distributions, symmetric and asymmetric ones.
Herbert Büning

On Nonparametric Tests for Trend Detection in Seasonal Time Series

We investigate nonparametric tests for identifying monotone trends in time series as they need weaker assumptions than parametric tests and are more flexible concerning the structure of the trend function. As seasonal effects can falsify the test results, modifications have been suggested which can handle also seasonal data. Diersen and Trenkler [5] propose a test procedure based on records and Hirsch et. al [8] develop a test based on Kendall's test for correlation. The same ideas can be applied to other nonparametric procedures for trend detection. All these procedures assume the observations to be independent. This assumption is often not fulfilled in time series analysis. We use the mentioned test procedures to analyse the time series of the temperature and the rainfall observed in Potsdam (Germany) from 1893 to 2008. As opposed to the rainfall time series, the temperature data show positive autocorrelation. Thus it is also of interest, how the several test procedures behave in case of autocorrelated data.
Oliver Morell, Roland Fried

Nonparametric Trend Tests for Right-Censored Survival Times

In clinical dose finding studies or preclinical carcinogenesis experiments survival times may arise in groups associated with ordered doses. Here interest may focus on detecting dose dependent trends in the underlying survival functions of the groups. So if a test is to be applied we are faced with an ordered alternative in the test problem, and therefore a trend test may be preferable. Several trend tests for survival data have already been introduced in the literature, e.g., the logrank test for trend, the one by Gehan [4] and Mantel [12], the one by Magel and Degges [11], and the modified ordered logrank test by Liu et al. [10], where the latter is shown to be a special case of the logrank test for trend. Due to their similarity to single contrast tests it is suspected that these tests are more powerful for certain trends than for others. The idea arises whether multiple contrast tests can lead to a better overall power and a more symmetric power over the alternative space. So based on the tests mentioned above two new multiple contrast tests are constructed. In order to compare the conventional with the new tests a simulation study was carried out. The study shows that the new tests preserve the nominal level satisfactory from a certain sample size but fail to conform the expectations in the power improvements.
Sandra Leissen, Uwe Ligges, Markus Neuhäuser, Ludwig A. Hothorn

Penalty Specialists Among Goalkeepers: A Nonparametric Bayesian Analysis of 44 Years of German Bundesliga

Penalty saving abilities are of major importance for a goalkeeper in modern football. However, statistical investigations of the performance of individual goalkeepers in penalties, leading to a ranking or a clustering of the keepers, are rare in the scientific literature. In this paper we will perform such an analysis based on all penalties in the German Bundesliga from 1963 to 2007. A challenge when analyzing such a data set is the fact that the counts of penalties for the different goalkeepers are highly imbalanced, leading to the question on how to compare goalkeepers who were involved in a disparate number of penalties. We will approach this issue by using Bayesian hierarchical random effects models. These models shrink the individual goalkeepers estimates towards an overall estimate with the degree of shrinkage depending on the amount of information that is available for each goalkeeper. The underlying random effects distribution will be modelled nonparametrically based on the Dirichlet process. Proceeding this way relaxes the assumptions underlying parametric random effect models and additionally allows to find clusters among the goalkeepers.
Björn Bornkamp, Arno Fritsch, Oliver Kuss, Katja Ickstadt

Permutation Tests for Validating Computer Experiments

Deterministic computer experiments are of increasing importance in many scientific and engineering fields. In this paper we focus on assessing the adequacy of computer experiments, i.e. we test if a computer experiment is predicting a corresponding real world phenomenon. A permutation test is presented which can be adapted to different situations in order to achieve good power.
Thomas Mühlenstädt, Ursula Gather

Parametric Inference

Exact and Generalized Confidence Intervals in the Common Mean Problem

Several exact confidence intervals for the common mean of independent normal populations have been proposed in the literature. Not all of these intervals always produce genuine intervals. In this paper, we consider three types of always genuine exact confidence intervals and compare these intervals with two known generalized confidence intervals for the common mean and a newly proposed one. Besides simulation results, two real data examples are presented illustrating the performance of the various procedures.
Joachim Hartung, Guido Knapp

Locally Optimal Tests of Independence for Archimedean Copula Families

A multivariate distribution can be decoupled into its marginal distributions and a copula function, a distribution function with uniform marginals. Copulas are well suited for modelling the dependence between multivariate random variables independent of their marginal distributions. Applications range from survival analysis over extreme value theory to econometrics. In recent years, copulas have attracted increased attention in financial statistics, in particular regarding modelling issues for high-dimensional problems like value-at-risk or portfolio credit risk. The well studied subclass of Archimedean copulas can be expressed as a function of a one-dimensional generating function ϕ. This class has become popular due to its richness in various distributional attributes providing flexibility in modelling. Here, we present locally optimal tests of independence for Archimedean copula families that are parameterized by a dependence parameter ϑ, where ϑ = 0 denotes independence of the marginal distributions. Under the general assumption of L 2 -differentiability at ϑ = 0 we calculate tangents of the underlying parametric families. For selected examples the optimal tests are calculated and connections to well-known correlation functionals are presented.
Jörg Rahnenführer

Design of Experiments and Analysis of Variance

Optimal Designs for Treatment-Control Comparisons in Microarray Experiments

Two-colour microarray experiments form an important tool in modern molecular biology. But they are very expensive and so careful planning of experiments is necessary. In this paper we determine optimal approximate designs for microarray experiments when only treatment-control comparisons are of interest. Based on these results we construct near optimal finite designs and compare their efficiencies with those of the corresponding star designs, which are often used in microarray experiments.
Joachim Kunert, R. J. Martin, Sabine Rothe

Improving Henderson's Method 3 Approach when Estimating Variance Components in a Two-way Mixed Linear Model

A two-way linear mixed model, consisting of three variance components, σ1 2, σ2 2 and σe 2 is considered. The variance component estimators are estimated using a well known non-iterative estimation procedure, Henderson's method 3. for σ2 1 we propose two modified estimators. The modification is carried out by perturbing the standard estimator, such that the obtained estimator is expected to perform better in terms of its mean square error.
Razaw al Sarraj, Dietrich von Rosen

Implications of Dimensionality on Measurement Reliability

We study some topics of the reliability of measurement, especially certain implications of multidimensionality and unidimensionality. We consider these aspects within a measurement framework focusing on one hand on the dimensionality of the measurement model and on the other hand on the dimensionality of the measurement scale. Working through theorems and examples we compare two reliability estimators, namely Cronbach's alpha and Tarkkonen's rho. It seems that there is not much use for Cronbach's alpha. It is based on unidimensional models and scales, while the models and scales used in practice are multidimensional. Tarkko-nen's rho seems to work well in multidimensional studies, giving support to the real purpose of reliability estimation which seems to have been lost for a quite long time.
Kimmo Vehkalahti, Simo Puntanen, Lauri Tarkkonen

Linear Models and Applied Econometrics

Robust Moment Based Estimation and Inference: The Generalized Cressie-Read Estimator

In this paper a range of information theoretic distance measures, based on Cressie-Read divergence, are combined with mean-zero estimating equations to provide an efficient basis for semi parametric estimation and testing. Asymptotic properties of the resulting semi parametric estimators are demonstrated and issues of implementation are considered.
Ron C. Mittelhammer, George G. Judge

More on the F-test under Nonspherical Disturbances

We show that the F-test can be both liberal and conservative in the context of a particular type of nonspherical behaviour induced by spatial autocorrelation, and that the conservative variant is more likely to occur for extreme values of the spatial autocorrelation parameter. In particular, it will wipe out the progressive one as the sample size increases.
Walter Krämer, Christoph Hanck

Optimal Estimation in a Linear Regression Model using Incomplete Prior Information

For the estimation of regression coefficients in a linear model when incomplete prior information is available, the optimal estimators in the classes of linear heterogeneous and linear homogeneous estimators are considered. As they involve some unknowns, they are operationalized by substituting unbiased estimators for the unknown quantities. The properties of resulting feasible estimators are analyzed and the effect of operationalization is studied. A comparison of the heterogeneous and homogeneous estimation techniques is also presented.
Helge Toutenburg, Shalabh, Christian Heumann

Minimum Description Length Model Selection in Gaussian Regression under Data Constraints

The normalized maximum likelihood (NML) formulation of the stochastic complexity Rissanen ([10]) contains two components: the maximized log likelihood and a component that may be interpreted as the parametric complexity of the model. The stochastic complexity for the data, relative to a suggested model, serves as a criterion for model selection. The calculation of the stochastic complexity can be considered as an implementation of the minimum description length principle (MDL) (cf. Rissanen [12]). To obtain an NML based model selection criterion for the Gaussian linear regression, Rissanen [11] constrains the data space appropriately. In this paper we demonstrate the effect of the data constraints on the selection criterion. In fact, we obtain various forms of the criterion by reformulating the shape of the data constraints. A special emphasis is placed on the performance of the criterion when collinearity is present in data.
Erkki P. Liski, Antti Liski

Self-exciting Extreme Value Models for Stock Market Crashes

We demonstrate the usefulness of Extreme value Theory (EVT) to evaluate magnitudes of stock market crashes and provide some extensions. A common practice in EVT is to compute either unconditional quantiles of the loss distribution or conditional methods linking GARCH models to EVT. Our approach combines self-exciting models for exceedances over a given threshold with a marked dependent process for estimating the tail of loss distributions. The corresponding models allow to adopt ex-ante estimation of two risk measures in different quantiles to assess the expected frequency of different crashes of important stock market indices. The paper concludes with a backtesting estimation of the magnitude of major stock market crashes in financial history from one day before an important crash until one year later. The results show that this approach provides better estimates of risk measures than the classical methods and is moreover able to use available data in a more efficient way.
Rodrigo Herrera, Bernhard Schipp

Consumption and Income: A Spectral Analysis

The relationship between aggregate income and consumption in the United Kingdom is analysed anew. This entails a close examination of the structure of the data, using a variety of spectral methods that depend on the concepts of Fourier analysis. It is found that fluctuations in the rate of growth of consumption tend to precede similar fluctuations in income, which contradicts a common supposition. The difficulty is emphasised of uncovering from the aggregate data a structural equation representing the behaviour of consumers.
D.S.G. Pollock

Stochastic Processes

Improved Estimation Strategy in Multi-Factor Vasicek Model

We consider simultaneous estimation of the drift parameters of multivari-ate Ornstein-Uhlebeck process. In this paper, we develop an improved estimation methodology for the drift parameters when homogeneity of several such parameters may hold. However, it is possible that the information regarding the equality of these parameters may not be accurate. In this context, we consider Stein-rule (or shrinkage) estimators to improve upon the performance of the classical maximum likelihood estimator (MLE). The relative dominance picture of the proposed estimators are explored and assessed under an asymptotic distributional quadratic risk criterion. For practical arguments, a simulation study is conducted which illustrates the behavior of the suggested method for small and moderate length of time observation period. More importantly, both analytical and simulation results indicate that estimators based on shrinkage principle not only give an excellent estimation accuracy but outperform the likelihood estimation uniformly.
S. Ejaz Ahmed, Sévérien Nkurunziza, Shuangzhe Liu

Bounds on Expected Coupling Times in a Markov Chain

In the author's paper “Coupling and Mixing Times in Markov Chains” (Res. Lett. Inf. Math. Sci, 11, 1–22, 2007) it was shown that it is very difficult to find explicit expressions for the expected time to coupling in a general Markov chain. In this paper simple upper and lower bounds are given for the expected time to coupling in a discrete time finite Markov chain. Extensions to the bounds under additional restrictive conditions are also given with detailed comparisons provided for two and three state chains.
Jeffrey J. Hunter

Multiple Self-decomposable Laws on Vector Spaces and on Groups: The Existence of Background Driving Processes

Following K. Urbanik, we define for simply connected nilpotent Lie groups G multiple self-decomposable laws as follows: For a fixed continuous one-parameter group (T t ) of automorphisms put \(L^{(0)} : = M^1 \left( G \right)\,and\,L^{(m + 1)} : = \{ \mu \in M^1 \left( G \right):\forall t > 0\ \exists\ v(t) \in L^{(m)} :\mu = T_t (\mu )*v(t)\} \,for \,m \ge 0.\)
Under suitable commutativity assumptions it is shown that also for m > 0 there exists a background driving Lévy process with corresponding continuous convolution semigroup (v s)s≥0 determining μ and vice versa. Precisely, μ and v s are related by iterated Lie Trotter formulae.
Wilfried Hazod

Matrix Algebra and Matrix Computations

Further Results on Samuelson's Inequality

In this paper we show that Samuelson's [11] inequality is essentially due to Gauss [6] whilst a more general result of the same type is due to Aitken [1, 2]. We also show that the adding-up constraint on the deviations from sample means implicit in Trenkler and Puntanen's [14] multivariate generalisation of Samuelson's Inequality can be regarded as a special case of a more general formulation involving a set of linear constraints on the deviations.
Richard William Farebrother

Revisitation of Generalized and Hypergeneralized Projectors

The notions of generalized and hypergeneralized projectors, introduced by Groβ and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474], attracted recently considerable attention. The list of publications devoted to them comprises now over ten positions, and the present paper briefly discusses some of the results available in the literature. Furthermore, several new characteristics of generalized and hypergeneralized projectors are established with the use of Corollary 6 in Hartwig and Spindelböck [Matrices for which A* and A† commute. Linear Multilinear Algebra 14 (1984) 241–256].
Oskar Maria Baksalary

On Singular Periodic Matrices

In this note we recall the concept of a singular periodic square matrix, admitting a positive integer power greater than one which is identical to the matrix itself. Characterizations involving the group inverse of a matrix are given and relationships with normal and EP matrices are investigated.
Jürgen Groß

Testing Numerical Methods Solving the Linear Least Squares Problem

The paper derives a general method for testing algorithms solving the Least-Squares-Problem (LS-Problem) of a linear equation system. This test method includes the generation of singular test matrices with arbitrary condition, full column rank and exactly representable generalized inverses, as well as a method for choosing general right hand sides. The method is applied to three LS-Problem solvers in order to assess under what conditions the error in the least squares solution is only linearly dependent on the condition number.
Claus Weihs

On the Computation of the Moore—Penrose Inverse of Matrices with Symbolic Elements

In this paper potential difficulties in using Greville's method for the computation of the Moore—Penrose inverse of a matrix that also contains symbolic elements are discussed. For the actual computation of the Moore—Penrose inverse of matrices whose elements are not numeric only, a Computer Algebra System has to be used. Initially, the computation of the Moore—Penrose inverse of a vector is considered which is a simple task if it only has numeric elements. If it contains symbolic elements, it might also be straightforward, but might turn out to be difficult. As Greville's method — an iterative algorithm that needs n steps for the computation of the Moore—Penrose inverse of an m by n matrix — requires the computation of the Moore—Penrose inverse of a vector in each step, the difficulty just mentioned might prevent the actual computation of the Moore—Penrose inverse of a matrix with symbolic elements.
Karsten Schmidt

On Permutations of Matrix Products

It is well-known that trace(AB) ≥ 0 for real-symmetric nonnegative definite matrices A and B. However, trace(ABC) can be positive, zero or negative, even when C is real-symmetric nonnegative definite. The genesis of the present investigation is consideration of a product of square matrices A =A 1 A 2A n. Permuting the factors of A leads to a different matrix product. We are interested in conditions under which the spectrum remains invariant. The main results are for square matrices over an arbitrary algebraically closed commutative field. The special case of real-symmetric, possibly nonnegative definite, matrices is also considered.
Hans Joachim Werner, Ingram Olkin

Special Topics

Some Comments on Fisher's α Index of Diversity and on the Kazwini Cosmography

Biodiversity, or biological diversity, is “the variety of life on our planet, measurable as the variety within species, between species, and the variety of ecosystems” [12, 41] and the most widely applied index of biodiversity is surely Fisher's α, defined implicitly by S = αloge{1+(n/α)}, where n is the number of individuals and S is the number of species. This index αwas first proposed over 60 years ago by R. A. Fisher in a three-part joint paper with A. Steven Corbet and C. B. Williams [14]. We also present some comments on the diversity of the paintings by Johannes Vermeer (1632–1675) depicted on postage stamps updating our findings in [3]. The earliest study of biodiversity seems to be that reported in the Kazwini Cosmography c. 1283; this study involved 72 different kinds of papillons that were collected in what we believe was Baghdad in c. 900 AD. We also found some magic squares in the Kazwini Cosmography. Our list of references is annotated and contains hyperlinks to open-access and restricted-access files on the internet.
Oskar Maria Baksalary, Ka Lok Chu, Simo Puntanen, George P.H. Styan

Ultimatum Games and Fuzzy Information

We consider the proposer's decision process in an ultimatum game where his uncertainty with respect to the responder's preferences and the associated acceptance threshold is modeled by a fuzzy set. Employing a three-step defuzzification strategy we determine the proposer's best possible claim which depends on his beliefs and his attitude towards risk. Furthermore, we derive an explicit solution for a specific class of fuzzy sets. From a more abstract point of view we analyze a game in which one player has a non-continuous objective function and where the uncertain point of discontinuity is determined by the other player's strategy.
Philip Sander, Peter Stahlecker

Are Bernstein's Examples on Independent Events Paradoxical?

Bernstein gave two examples showing that a collection of pairwise independent random events need not to be jointly independent. These examples were numbered by Stoyanov among the most fascinating counterexamples in probability. Considering the minimal sample size for existing n independent and pairwise independent but jointly dependent random events we reveal the fact that the second situation is more often. In consequence it is rather a rule than a paradox.
Czesław Stępniak, Tomasz Owsiany

A Classroom Example to Demonstrate Statistical Concepts

A hands-on example to alleviate the transition process from probability models to statistical inference is proposed which can be used in a statistics lecture. It is very easy to grasp and has the benefit that many skills learned so far can be applied. Concepts like parameter estimation, confidence intervals and testing are addressed in this context. It can serve as reference when more abstract concepts such as unbiasedness, mean square error, pivot quantities, confidence level or p-values are treated later on in the course.
Dietrich Trenkler


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